Above as expected answers to your questions. Let me try to give you some more depth to being in LP and hiking.
First of all - is you are "list" hikers - consider leaving Whiteface for your final as there is an autoroad to the top that non-hikers can use to help you celebrate on top.
Second - Cascade/Porter from Rt. 73 is the most common first hike among ADK hikers. Great hike - but what I don't like is that it may be crowed and it is not a "typical" ADK hike (ie: walk into a trailhead and then start acsent - see wilderness).
Wright/Algonquin is classis - adding Iroquois is optional. Know that you will pay to park at the ADK Loj parking area and that there are plenty of hikers and wilderness users around (plan on seeing activity and finding many autos). However, there are many trails that are serviced out of this area and crowds are dispersed pretty well once you are out on the trail (you can find solitude in the ADK's but this is not the place). Wright's summit is one of my favorites!
You want to make it an awesome hike - and see lots... go into Marcy Dam and follow the trail through Avalance Pass, over the Hitch 'em ups and acsent via the Avalanche Mtn trail... reportedly one of the steepest in the ADKs but I wouldn't agree with that - I loved that trail. I comes out on a col between Iroquois and AlG - and you come down over AlG and out - hitting Wright if you feel you have time.
Agree Colden is one of the best and you'll see plenty. Going up through Lake Arnold is most common route. Options for loops do exist.
Closer shorter is Phelps - not an awesome summit but nice views and a nice ADK feel. (Out ot Loj area).
Giant is an ADK classis -and spending time at Roaring Brook is the best (Rt 73 area). Suggestion for Big Slide is also an excellent suggestion -- this can be a loop as Brothers - Johns Brook (remember that parking in this area is in the "Garden Parking Lot" and space is limited - and fall leaf time will fill it).
Finally - LP is a fun and exciting place. There is so much to do in the area that if this is your first trip - you'll want to spend time in town and in the surrounding area seeing sights, shopping and eating. Our group always hiked on one day and took the second day for a "shopping day" or for others - a "lighter" hiking day.
Lighter day hikes includes Mt. Jo (Loj area), Baxter Mtn, Sunrise Trail to Mt. Gilligan (New Russia area), Round Mtn or Roaring Brook by itself (off Rt 73) and many more.
Driving Times from LP: Loj - abt 20/25 mins. To Garden and Rt 73 Trailheads - abt 40/45 min.
Okay - now finally - The Cottage is a fine place for mid-day cocktails, evening coffee and afternoon porch snacks (on Mirror Lake). On the other end of the lake is LP Brewery - a must stop for anyone who posts on this site!!
Let us know more as you fine-tune your plans. You're going to love this place!!
